Solve the Inequality for x (x-1)^2-7>(x-2)^2

${(x - 1)}^{2} - 7 > {(x - 2)}^{2}$

Add $7$ to both sides of the inequality.

${(x - 1)}^{2} > {(x - 2)}^{2} + 7$

Subtract ${(x - 2)}^{2}$ from both sides of the inequality.

${(x - 1)}^{2} - {(x - 2)}^{2} > 7$

Subtract $7$ from both sides of the inequality.

${(x - 1)}^{2} - {(x - 2)}^{2} - 7 > 0$

Simplify ${(x - 1)}^{2} - {(x - 2)}^{2} - 7$ .

Simplify each term.

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$(x - 1) (x - 1) - {(x - 2)}^{2} - 7 > 0$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$x (x - 1) - 1 (x - 1) - {(x - 2)}^{2} - 7 > 0$

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 1 (x - 1) - {(x - 2)}^{2} - 7 > 0$

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - {(x - 2)}^{2} - 7 > 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 - {(x - 2)}^{2} - 7 > 0$

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 - {(x - 2)}^{2} - 7 > 0$

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - 1 x - 1 \cdot - 1 - {(x - 2)}^{2} - 7 > 0$

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - x - 1 \cdot - 1 - {(x - 2)}^{2} - 7 > 0$

Multiply $- 1$ by $- 1$ .

$x^{2} - x - x + 1 - {(x - 2)}^{2} - 7 > 0$

Subtract $x$ from $- x$ .

$x^{2} - 2 x + 1 - {(x - 2)}^{2} - 7 > 0$

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$x^{2} - 2 x + 1 - ((x - 2) (x - 2)) - 7 > 0$

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$x^{2} - 2 x + 1 - (x (x - 2) - 2 (x - 2)) - 7 > 0$

Apply the distributive property.

$x^{2} - 2 x + 1 - (x \cdot x + x \cdot - 2 - 2 (x - 2)) - 7 > 0$

Apply the distributive property.

$x^{2} - 2 x + 1 - (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) - 7 > 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} - 2 x + 1 - (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) - 7 > 0$

Move $- 2$ to the left of $x$ .

$x^{2} - 2 x + 1 - (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) - 7 > 0$

Multiply $- 2$ by $- 2$ .

$x^{2} - 2 x + 1 - (x^{2} - 2 x - 2 x + 4) - 7 > 0$

Subtract $2 x$ from $- 2 x$ .

$x^{2} - 2 x + 1 - (x^{2} - 4 x + 4) - 7 > 0$

Apply the distributive property.

$x^{2} - 2 x + 1 - x^{2} - (- 4 x) - 1 \cdot 4 - 7 > 0$

Simplify.

Multiply $- 4$ by $- 1$ .

$x^{2} - 2 x + 1 - x^{2} + 4 x - 1 \cdot 4 - 7 > 0$

Multiply $- 1$ by $4$ .

$x^{2} - 2 x + 1 - x^{2} + 4 x - 4 - 7 > 0$

Combine the opposite terms in $x^{2} - 2 x + 1 - x^{2} + 4 x - 4 - 7$ .

Subtract $x^{2}$ from $x^{2}$ .

$- 2 x + 1 + 0 + 4 x - 4 - 7 > 0$

Add $- 2 x + 1$ and $0$ .

$- 2 x + 1 + 4 x - 4 - 7 > 0$

Add $- 2 x$ and $4 x$ .

$2 x + 1 - 4 - 7 > 0$

Subtract $4$ from $1$ .

$2 x - 3 - 7 > 0$

Subtract $7$ from $- 3$ .

$2 x - 10 > 0$

Factor $2$ out of $2 x - 10$ .

Factor $2$ out of $2 x$ .

$2 (x) - 10 > 0$

Factor $2$ out of $- 10$ .

$2 x + 2 \cdot - 5 > 0$

Factor $2$ out of $2 x + 2 \cdot - 5$ .

$2 (x - 5) > 0$

Divide each term in $2 (x - 5) > 0$ by $2$ and simplify.

Divide each term in $2 (x - 5) > 0$ by $2$ .

$\frac{2 (x - 5)}{2} > \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (x - 5)}{2} > \frac{0}{2}$

Divide $x - 5$ by $1$ .

$x - 5 > \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x - 5 > 0$

Add $5$ to both sides of the inequality.

$x > 5$

The result can be shown in multiple forms.

Inequality Form:

$x > 5$

Interval Notation:

$(5, \infty)$

Solve the Inequality for x (x-1)^2-7>(x-2)^2

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